"I urge Egypt, the people and leadership, to seek with us the success of this (reconciliation) move as part of Arab solidarity," the king said, quoted by the official SPA news agency.
The agreement between Gulf states reached in Riyadh on Sunday offers "a general framework for unity, consensus, and an end to differences" between Arab states, the king said.
The ouster of Egypt's Islamist president Mohamed Morsi last year was at the core of differences among Gulf states.
Tensions between the four Gulf states hit a new low in March when Bahrain, Saudi and the UAE announced the withdrawal of their envoys from Qatar in protest at its "interference" in their internal affairs by supporting Islamists.
On Sunday, Gulf leaders agreed to return their ambassadors to Doha, signalling an end to the dispute.
In the statement, King Abdullah said the kingdom looked forward "to the start of a new phase of consensus" among Arab nations.
Sunday's reconciliation comes amid security fears over the rise of the Islamic State jihadist group in Syria and Iraq.
Bahrain, Kuwait, Qatar, Saudi Arabia and the UAE have all joined US-led air strikes against the IS in Syria. Oman is the only Gulf Cooperation Council member not to have done so.
